A set of integers
${\displaystyle A=\{a_{1},a_{2},...,a_{m}\}\quad a_{1}<a_{2}<...<a_{m}} $ is a Golomb ruler if and only if
${\displaystyle \forall i,j,k,l\in \left\{1,2,...,m\right\},a_{i}-a_{j}=a_{k}-a_{l}\land i\neq j\iff i=k\land j=l.}$
Essentially for all $i,j$ we have $a_i-a_j$ distinct.
Take $z_1,\dots,z_t\in\mathbb Z$ with condition $0<|z_i|\leq|z_{i+1}|$ at every $i$.
$(-1,1,2)$, $(-2,-3,4)$ and $(-3,4,5)$ are valid weights.
$(-2,-3,1)$ and $(-2,-1)$ are invalid.
Define $(z_1,\dots,z_t)$ weighted Golumb unrestricted ruler to be set of $\{a_1,\dots,a_m\}$ with possibly $a_i\leq{a_{i+1}}$ at every $i$ such that each of $\sum_{j=1}^tz_ja_{i_j}$ is distinct where $i_j\neq i_{j+1}$ at every $j$ where weights are with above condition.
A regular Golumb (which is not unrestricted) ruler is $z_1=-1$ and $z_2=1$.
Take weights to be $(-1,1^\ell,\dots,t^\ell)$ where $\ell\in\mathbb N_{>0}$ is fixed.
What is the minimum length unrestricted ruler to obtain $d$ distinct integers with weights $(-1,1^\ell,\dots,t^\ell)$?
Has this been studied anywhere?
